Design, fabrication, characterization and reliability study of CMOS-MEMS Lorentz-force magnetometers

This article presents several design techniques to fabricate micro-electro-mechanical systems (MEMS) using standard complementary metal-oxide semiconductor (CMOS) processes. They were applied to fabricate high yield CMOS-MEMS shielded Lorentz-force magnetometers (LFM). The multilayered metals and oxides of the back-end-of-line (BEOL), normally used for electronic routing, comprise the structural part of the MEMS. The most important fabrication challenges, modeling approaches and design solutions are discussed. Equations that predict the Q factor, sensitivity, Brownian noise and resonant frequency as a function of temperature, gas pressure and design parameters are presented and validated in characterization tests. A number of the fabricated magnetometers were packaged into Quad Flat No-leads (QFN) packages. We show this process can achieve yields above 95 % when the proper design techniques are adopted. Despite CMOS not being a process for MEMS manufacturing, estimated performance (sensitivity and noise level) is similar or superior to current commercial magnetometers and others built with MEMS processes. Additionally, typical offsets present in Lorentz-force magnetometers were prevented with a shielding electrode, whose efficiency is quantified. Finally, several reliability test results are presented, which demonstrate the robustness against high temperatures, magnetic fields and acceleration shocks.

= (Air damping) (1) where is a proportionality parameter that depends on the considered geometry and is the resonance frequency. The Q dependency with pressure can be introduced using a pressure-dependent artificial viscosity ( ), which has been studied for different cases (squeezed-film or shear flow, molecular or slip-flow regime, diffuse or specular gas particle reflections...) [4][5][6][7][8] . All these approximations have a common form, which is: where and are two free parameters, 0 is the dynamic viscosity of the gas at a specified temperature (1.81 × 10 −5 Pa s at 300 K and ambient pressure), and is the Knudsen number:    According to the Kinetic theory of gases is proportional to ∕ . Remarkably, the artificial viscosity approach works reasonably well even when > 1 and therefore the dissipation is caused not by viscous forces but by the impact of noninteracting gas molecules. This is called the ballistic or free molecular flow regime.
Intrinsic Damping: Intrinsic damping generally represents the Q factor upper limit at sufficiently low pressure. It arises from relaxation loss mechanisms within the resonating structure itself 2 . The better known example may be thermoelastic damping (TED), which is an absolute lower bound on intrinsic damping, but friction loss mechanisms like surface loss or phase boundary slipping in multilayer structures should also be considered in CMOS-MEMS structures. Friction loss mechanisms are a ubiquitous phenomenon and, along with TED, are best described by the Zener's anelastic relaxation theory 9,10 . In this theory, the Q factor resultant from intrinsic damping mechanisms would be given by: where Δ is the relaxation strength and = 1∕(2 ) is the Debye frequency associated with the relaxation time of the ith mechanism ( ). Generally, one mechanism is dominant and it is sufficient to consider = 1. The minimum Q factor occurs when the vibration is at the Debye frequency of the dominant one. Depending on whether is well *Sensing electrode is hollow between layers and air can ow through. **Double vertical gap of , one of them with holes and air can ow through. Table 1 Parameters used in Eq. (7).
below (isothermal regime) or well above it (adiabatic regime) the Q dependency with is the opposite: In the case of TED and for uniform beams 2 : where is the Young's Modulus, the thermal expansion coefficient, the density, the specific heat, the thermal conduction coefficient and the beam thickness.
Q factor characterization: Our data clearly shows a ∝ −1 dependency in the ballistic regime, which implies = 1 in Eq. (2), close to most formulas in Veijola et al. 6 , Li and Hughes 8 . Most authors use Veijola's formula with = 1.159, intended for diffusely rejecting identical surfaces 6 , but it does not work well in our case (3 − 5 µm wide and 100 − 800 µm long BEOL CMOS beams with 0.35 − 1.00 µm gaps where both slide and squeeze film damping take place).
The value models ( ) in the fluidic regime ( ≤ 1). Typically, it may range from = 2 for shear flow 5 to values not usually higher than 10, as shown in Li and Hughes 8 . In our case, = 5 worked reasonably well. The proportionality factor between Q and ∕ in Eq. (1) defines the slope of the curve in the ballistic regime. It turns out to be = 6.80 × 10 −11 Pa s 2 for the z devices and = 3.50 × 10 −11 Pa s 2 for the vertical devices.
The shortest devices showed Q factors up to 30 % higher than initially expected in the air-damped region, according to their resonance frequency and Eq. (1). We think it may be caused by the air not being able to escape from the closing gap fast enough and starting to behave more like a spring and less like a damper. In this case, the damping coefficient will change with frequency as ∝ 1∕(1 + 2 ∕ 2 ), where is the cut-off frequency 11 . The approximate cut-off frequency that best fitted the data was ≈ 2.3 MHz for the z device and ≈ 3.0 MHz for the vertical device.
After adding all the discussed corrections to Eq. (1), the Q factor due to air damping is, finally: where the pressure dependence is contained in ∝ ∕ and 0 = ( ) may be considered independent of . Table 1 summarizes the parameters used for the 2 cases represented in Fig. 1 (highlighted), and also provides experimental data for three additional cases. At low pressures, another damping mechanism becomes the dominant one and the measured quality factors reach a plateau (see Fig. 1). We have plotted the measured Q factor as a function of the resonant frequency at 1 µbar in Fig. 2 in order to analyze the dominant damping mechanism. The z devices (see plateaus in Fig. 1a and circular data points in Fig. 2) operate in the isothermal region, where ∝ 1∕ , just the inverse proportionality of that found in the  Table 2 Quality factors due to TED ( ) used in Eq. (8).
air-damped region. On the other hand, the shorter xy devices ( Fig. 1b and square data points around 1MHz in Fig. 2) seem to operate close to their Debye frequency given than the Q factor does not depend that much on the vibration frequency. Duwel et al. 12 has shown that TED is an important loss mechanism for flexural modes. However, Prabhakar and Vengallatorer observed in Prabhakar and Vengallatore 13 that internal friction is much higher than TED when < 1 MHz in some bilayer structures. Given that CMOS-MEMS devices are multilayered and more complex than theirs, intrinsic friction losses might be important. However, finite element analysis (FEA) carried out by us predicted Q factors due to TED very similar to the measured ones for both types of devices. The simulated TED Debye frequency for the vertical device (0.8 MHz) is substantially smaller than for the lateral one (larger than 2 MHz). This explains the higher Q factors for the longest devices at low pressure. However, all the CMOS BEOL layers must be included in the FEA model (see cross-sections in Fig. 2), even the adhesion and antireflective coatings (Titanium and Titanium Nitride) in order to perform sufficiently accurate predictions. These layers play an important role because they have a low thermal conductance which decreases the associated TED Debye frequency and this determines greatly the simulated Q factor. Also, stress in the beams was included in the simulations given its importance in highly stressed structures 2,14 . The only deviation from simulations takes place in the shortest lateral devices, for which the TED Q factor is overestimated. It might be evidence of another damping mechanism that we have not identified.
The total Q factor is, therefore: All the solid lines in Fig. 1 were generated with Eq. (8). The value used for is shown in Table 2 and was deduced from the measured data, rather than the simulation because, as already mentioned, there is some disagreement for the shortest lateral devices at 1 µbar. With that exception, Eq. (8) fits very well the measured data.